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Let $f : \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$. I believe the following inequality holds, but I am not sure how to prove it: $$ |\sup _x f(x) - \sup _y g(y)| \leq | \sup _x ( f(x) - g(x) ) | $$

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The statement is false, and it is relatively easy to find a counterexample. I was considering the lemma for vectors, i.e., $x, y \in \mathbb R ^d$, and in this case it clearly fails. Let $i ^* \in \arg\max _{i \in [d]} x _i$, $j ^* \in \arg\max _{j \in [d] } y _j$, and $k ^* \in \max _{k \in [d]} ( x _k - y _k )$.

Split the proof in two cases; if $x _{i ^*} - y _{j ^*} \geq 0$, the statement is true; we have $ x _{k ^*} - y _{k ^*} \geq x _{i ^*} - y _{i ^*} \geq x _{i ^*} - y _{j ^*} \geq 0$, proving the result. If $x _{i ^*} - y _{j ^*} < 0$, there is a simple counterexample. Let $d = 2$, and take $x = (0, 0)$ and $y = (1, 0)$, then $|x _{i ^*} - y _{j *}| = 1$, but $| x _{k ^*} - y _{k ^*} | = 0$.